Physics calculator

Cantilever Tip Deflection Calculator

Free-end deflection of a cantilever with an intermediate point load. δ = P a² (3L − a) / (6 E I). Runs locally.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance

Input interpretation: Enter values to calculate.
Result: —
Assurance: Engineering
Declared partition coverage: PASS · 2/2 declared partitions (domain, invalid-domain) · Matrix
Known limitations: Physics L1 gap capability
Core CVP does not include live graph, viewport, or pointer interaction.
Model: Free-end deflection of a cantilever with one concentrated load between the fixed end and the tip.
Scope: Calculation runs locally in the browser; values are not uploaded.
Verification: Engine tested · Source checked · v1.0.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed
CVP identity: 0/0 property · digest d0a478e1cef0
Legacy regression: 3/3 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status: STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 1/1 golden · 1/1 CVP boundary · 1/1 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources: Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
Sources
Evidence: 2 legacy golden · 1 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 1/1 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Free-end deflectionδ = P a² (3L − a) / (6 E I)

ia = L recovers δ = P L³ / (3 E I). The deflection at the load is a different page.

How to use

Enter the load, its distance from the fixed end, the length, the modulus, and the area moment

L, E, and I must be positive. a must be greater than 0 and no greater than L.

Read the free-end deflection

The sign of P is the sign of δ.

Example calculations

Common configurations with formula and result.

Load at mid-length

P = 48, a = 1, L = 2, E = 1, I = 1

δ = 48 × 1² × (6 − 1) / 6

δ = 40

Load at the free end

P = 48, a = 2, L = 2, E = 1, I = 1

δ = P L³ / (3 E I)

δ = 128

Cantilever Tip Deflection calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 3/3 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: A cantilever of length L is fixed at one end and carries one concentrated load P at distance a from the fixed end. The deflection at the free end is δ = P a² (3L − a) / (6 E I).
What it calculates: Free-end deflection of a cantilever with one concentrated load between the fixed end and the tip.
Inputs: P
a
L
E
I
Outputs: delta
Formula: δ = P a² (3L − a) / (6 E I).
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
a = L recovers δ = P L³ / (3 E I). The deflection at the load is a different page.
Units: P, a, L, E, and I set the deflection unit.
Boundary conditions: missing P, a, L, E, or I → MISSING_REQUIRED_INPUT
L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
a ≤ 0 or a > L → INVALID_INPUT
Example: P=48 a=1 L=2 E=1 I=1 → δ=40
Validation cases: 3 published on this page · 3/3 tests · Production surface contract 6/6 · View evidence
P=48 a=1 L=2 E=1 I=1 → δ=40
P=48 a=2 L=2 E=1 I=1 → δ=128
a=0 → INVALID_INPUT
Sources: Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
Supports: The free-end deflection is P a² (3L − a) / (6 E I).
Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
Supports: Supports the page formula: δ = P a² (3L − a) / (6 E I) at the free end.
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the free-end deflection of a cantilever when the concentrated load is not necessarily at the tip.

Supported and not supported

Supported: δ = P a² (3L − a) / (6 E I). a = L recovers the tip-load page.

Not supported: the deflection at the load when a is shorter than L, a simply supported span, and a uniform load. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.cantilever_tip_deflection · tool id: cantilever-tip-deflection · pin 1.0.0.

{ "P": 48, "a": 1, "L": 2, "E": 1, "I": 1 }

Other calculators in this family: Angled Pull Calculator, Angled Pull on an Incline Calculator, Area Moment Calculator, Beam Deflection Calculator, Beam Moment Deflection Calculator, Beam Moment Maximum Deflection Calculator, Bending Stress Calculator, Cantilever Deflection Calculator . Explore all Statics & Strength.

Frequently asked questions

Key distinctions behind the calculation.

Why is the tip farther than the load?

Past the load the beam is straight, at the slope it already has. For a = 1 and L = 2 the tip moves 40 while the load moves 16.